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The Derivative

Problems

Problems

Since the derivative of a function
f
at any value
x0
where the function is
differentiable is just a number,
f'(x0)
, we may define the derivative function,
denoted
f'(x)
or
(x)
, to be the function that maps a value
x
where
f
is differentiable to the derivative of
f
at
x
. Since
f'
is a function in
its own right, we can graph it, just like any other function.

For example, let us calculate the derivative function (also called simply the
"derivative") of
f (x) = 5x
. Since the graph of this function is a line with slope
5
, we might guess that the derivative will equal
5
at every point. We verify this
with the formula for the
derivative of
f
at a point
x
: